Newsgroups: comp.lang.apl
Path: watmath!watserv2.uwaterloo.ca!torn!cs.utexas.edu!uunet!s5!sethb
From: sethb@fid.morgan.com (Seth Breidbart)
Subject: Re: Numerical Analysis + APL/J
Message-ID: <1992Sep1.043755.27813@fid.morgan.com>
Organization: my opinions only
References: <1184@kepler1.rentec.com> <1992Aug27.145141.22803@fid.morgan.com> <1222@kepler1.rentec.com>
Date: Tue, 1 Sep 1992 04:37:55 GMT
Lines: 61

In article <1222@kepler1.rentec.com> andrew@rentec.com (Andrew Mullhaupt) writes:
>In article <1992Aug27.145141.22803@fid.morgan.com> sethb@fid.morgan.com (Seth Breidbart) writes:
>>The problem is "Find a group multiplication table with some other
>>property".  I, the oracle, happen to know that addition mod n has that
>>property; you don't.
>
>This is not a sensible problem. How can I check if the table has a property
>if I don't know what it is?

You know what the property is, and you know how to check for it.  You
just don't know that addition mod n has that property.  Remember, you
specified that a solution is given by an oracle.

> But this is irrelevant. If you consider that we
>are trying to show that I can check a property as fast as you can generate
>the general case. If you don't generate the general case, you still end up
>with a lower bound, but not a very good one.

I claim to have a method to generate groups faster than you can check
that something is a group.  My method might not generate all groups,
but with a bit of randomness, it can generate groups that you can't
check as fast as I can generate them.

>>My claim was that it is sometimes easier to generate an object with a
>>given property than to prove that a specific (general) object has that
>>property. 
>
>It might seem easier, but it doesn't have anything to do with whether you
>can get lower bounds.

If I can construct an object faster than you can check it, then your
checking is obviously not a lower bound on my construction.

>>A better example might be "Produce a 200-digit number that
>>is the product of two 100-digit primes".  It's not that hard to find
>>100-digit primes and multiply them, but if I only give you the
>>200-digit number, it's very hard (we believe) to determine whether it
>>is the product of two 100-digit primes.
>
>The oracle tells you the primes and you check by multiplying them.

What oracle?

> A fine
>lower bound. In fact, there might be some clever way to speed up the
>multiplication of two primes, but so what. Note that it is totally believable
>that however long the multiplication takes is a lower bound for checking the
>given property. What is dubious is that it is a very sharp lower bound...

Now you're demanding an oracle that will tell you whatever you want.
The original problem had an oracle that will tell you the answer, but
not how to get it.

>Not at all. It's useful lower bounds which are hard to construct. We get lucky
>in the QR case.

I still don't believe your "proof".  Some things are easier to
construct than they are to check, unless you have the full
construction available.

Seth		sethb@fid.morgan.com
